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I was thinking about the impending collision between M31 and the milky way galaxy, and I've heard that since a galaxy is mostly space most of the contents of both galaxies would be just fine. Could a similar effect be responsible for the late heavy bombardment era? Just instead of galaxies, stellar systems?

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There are several potential explanation for the cause of the late heavy bombardment. One of them includes a passing star that disturbed the Oort cloud. A "stellar system collision" would have destroyed the solar system, though. While galaxies are mostly empty and a galactic collision won't involve that many actual collisions (if any), the gravitational interactions between the galaxies actually changes star orbits around the galactic centers. Similarly a "stellar system collision" would heavily change planetary orbits, even if there were no collisions at first.

Another hypothesis is a totally "interior" one: A 2:1 orbital resonance between Jupiter and Saturn changed the orbits of Neptune and Uranus, potentially even causing them to swap positions. Neptune's new orbit then affected many Kuiper belt objects, either sending them way out to form or join the Oort cloud, or sending them in as part of the late heavy bombardment.

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Recent studies and simulations reported in the paper Impact-induced compositional variations on Mercury (Edgard et al. 2014), suggest that the Late Heavy Bombardment (LHB) was

triggered by giant planet migration during which the main belt and E-belt are excited to higher impact velocities when resonances with the giant planets sweep across the primordial asteroid belt. These simulations indicate the E-belt is the principal source of LHB impactors.

The 'E-belt' is a simulated and theorised primordial extension of the asteroid belt that extended

the primordial asteroid belt could have originally extended well into the Mars-crossing zone

In the paper The Primordial Excitation and Clearing of the Asteroid Belt (Petit et al. 2001), state an important point to be considered when answering this question:

most of the asteroids we see today are not primordial, but fragments of larger asteroids destroyed in a collision. Only the largest asteroids retain characteristics that relate to the formation of the asteroid belt and were not drastically changed by the later evolution.

In their simulations, they suggest that the asteroid belt could have been a major feature of the entire inner solar system where the 'planetary embryos are forming (from 0.5 to 4AU), stating:

the presence of large embryos in the inner Solar System for about 100 to 200 My after Jupiter has reached its present-day mass provides an efficient mechanism for depleting the asteroid belt of most of its primordial mass and for dynamically exciting the remaining small bodies.

Related to the Late Heavy Bombardment:

few percent of the particles end up on these long-lived orbits, equivalent to several times the mass of the present asteroid belt. These orbits are unstable on a long timescale and represent a potential source of impactors for the Late Heavy Bombardment

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If a star was ping-pong ball size, even the dense cores of colliding galaxies would be like ping-pong balls flying past, 540 miles apart.

A star is about 5 light-seconds across and the average separation between stars is about 6 light years, or about 220 million light seconds. So stars make up about 1 part in 2.5e15 of the galactic plane, and the odds of any two arbitrary stars colliding when galaxies pass through each other are about 1 in 250 quadrillion.

Assuming the galactic centers of mass pass through each other, the odds of any of 2 trillion stars colliding, with each pair having that probability, is 1 in 125,000.

Only 1 in 125,000 galaxy collisions sees even a single pair of stars touch (modulo tidal effects).

The moon was pummeled in the bombardment. The earth would look that beat up were it not for erosion. But stars don't collide because their sizes are too small compared to the vast interstellar distances involved.

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